Firstly, you should really never install VS on a production server. There is no reason to do this. It could negatively impact the server, requires you keep it up to date and is a potential access point for malicious software. So my recommendation is you remove VS from the SQL server.
As for the target version, I don't think that is the issue. SQL 2022 can run SSIS packages targeting SQL 2019. All you need to do is deploy the SSIS package to the SQL 2022 instance and it should run. Did you actually deploy the SSIS package to your server or are you just trying to run it manually? The easiest way to deploy it is to open it in VS, right click the package in Solution Explorer and select the option to Deploy. It should deploy everything needed for that SSIS package to run.
Once you've deployed then run SSMS, go to the SSISDB database and find the SSIS package. Then execute the package. If it runs then everything is deployed properly.
Things to consider here:
- If the SSIS package is using shared data sources then you'd need to ensure that the appropriate data sources are also deployed.
- If you are using runtime configuration files (parameter files I think they are called) then you need to ensure they are deployed as well.
- If you're running into errors then enable diagnostic logging on the package on the server so you can get to the actual errors.
- If you rely on any third party components in SSIS then they have to be deployed first. This is outside the package deployment.
- Check to see what bitness your SSIS package is running under. Ensure that everything you rely on supports that bitness. I think VS uses 64-bit by default, but I could be wrong.
To develop SSIS packages going forward you should install VS 2022 on a development server. Copy your SSIS package to it and develop against that. Note that you can still target your SQL 2022 instance running wherever, that's the purpose of the Connection Manager. Once you have your package ready to go then, as mentioned earlier, you can have VS deploy to the SQL server for production use. Alternatively you can use a CLI or SSMS to deploy the package by hand.